Arens–Fort space

In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, [Jr.]

Definition

The Arens–Fort space is the topological space where is the set of ordered pairs of non-negative integers A subset is open, that is, belongs to if and only if:

does not contain or
contains and also all but a finite number of points of all but a finite number of columns, where a column is a set with fixed.

In other words, an open set is only "allowed" to contain if only a finite number of its columns contain significant gaps, where a gap in a column is significant if it omits an infinite number of points.

Properties

It is

Hausdorff
regular
normal

It is not:

second-countable
first-countable
metrizable
compact
sequential
Fréchet–Urysohn

There is no sequence in that converges to However, there is a sequence in such that is a cluster point of